(-1)-enumeration of plane partitions with complementation symmetry

We compute the weighted enumeration of plane partitions contained in a given box with complementation symmetry where adding one half of an orbit of cubes and removing the other half of the orbit changes the weight by -1 as proposed by Kuperberg. We use nonintersecting lattice path families to accomplish this for transpose-complementary, cyclically symmetric transpose-complementary and totally symmetric self-complementary plane partitions. For symmetric transpose-complementary and self-complementary plane partitions we get partial results. We also describe Kuperberg's proof for the case of cyclically symmetric self-complementary plane partitions.Comment: 41 pages, AmS-LaTeX, uses TeXDraw; reference adde

Self-complementary plane partitions by Proctor's minuscule method

A method of Proctor [European J. Combin. 5 (1984), no. 4, 331-350] realizes the set of arbitrary plane partitions in a box and the set of symmetric plane partitions as bases of linear representations of Lie groups. We extend this method by realizing transposition and complementation of plane partitions as natural linear transformations of the representations, thereby enumerating symmetric plane partitions, self-complementary plane partitions, and transpose-complement plane partitions in a new way

Nullity and Loop Complementation for Delta-Matroids

We show that the symmetric difference distance measure for set systems, and more specifically for delta-matroids, corresponds to the notion of nullity for symmetric and skew-symmetric matrices. In particular, as graphs (i.e., symmetric matrices over GF(2)) may be seen as a special class of delta-matroids, this distance measure generalizes the notion of nullity in this case. We characterize delta-matroids in terms of equicardinality of minimal sets with respect to inclusion (in addition we obtain similar characterizations for matroids). In this way, we find that, e.g., the delta-matroids obtained after loop complementation and after pivot on a single element together with the original delta-matroid fulfill the property that two of them have equal "null space" while the third has a larger dimension.Comment: Changes w.r.t. v4: different style, Section 8 is extended, and in addition a few small changes are made in the rest of the paper. 15 pages, no figure

The Group Structure of Pivot and Loop Complementation on Graphs and Set Systems

We study the interplay between principal pivot transform (pivot) and loop complementation for graphs. This is done by generalizing loop complementation (in addition to pivot) to set systems. We show that the operations together, when restricted to single vertices, form the permutation group S_3. This leads, e.g., to a normal form for sequences of pivots and loop complementation on graphs. The results have consequences for the operations of local complementation and edge complementation on simple graphs: an alternative proof of a classic result involving local and edge complementation is obtained, and the effect of sequences of local complementations on simple graphs is characterized.Comment: 21 pages, 7 figures, significant additions w.r.t. v3 are Thm 7 and Remark 2

Complementation, Local Complementation, and Switching in Binary Matroids

In 2004, Ehrenfeucht, Harju, and Rozenberg showed that any graph on a vertex set $V$ can be obtained from a complete graph on $V$ via a sequence of the operations of complementation, switching edges and non-edges at a vertex, and local complementation. The last operation involves taking the complement in the neighbourhood of a vertex. In this paper, we consider natural generalizations of these operations for binary matroids and explore their behaviour. We characterize all binary matroids obtainable from the binary projective geometry of rank $r$ under the operations of complementation and switching. Moreover, we show that not all binary matroids of rank at most $r$ can be obtained from a projective geometry of rank $r$ via a sequence of the three generalized operations. We introduce a fourth operation and show that, with this additional operation, we are able to obtain all binary matroids.Comment: Fixed an error in the proof of Theorem 5.3. Adv. in Appl. Math. (2020

Quaternary matroids are vf-safe

Binary delta-matroids are closed under vertex flips, which consist of the natural operations of twist and loop complementation. In this note we provide an extension of this result from GF(2) to GF(4). As a consequence, quaternary matroids are "safe" under vertex flips (vf-safe for short). As an application, we find that the matroid of a bicycle space of a quaternary matroid is independent of the chosen representation. This extends a result of Vertigan [J. Comb. Theory B (1998)] concerning the bicycle dimension of quaternary matroids.Comment: 8 pages, no figures, the contents of this paper is now merged into v2 of [arXiv:1210.7718] (except for this comment, v2 is identical to v1

Property lattices for independent quantum systems

We consider the description of two independent quantum systems by a complete atomistic ortho-lattice (cao-lattice) L. It is known that since the two systems are independent, no Hilbert space description is possible, i.e. $L\ne P(H)$, the lattice of closed subspaces of a Hilbert space (theorem 1). We impose five conditions on L. Four of them are shown to be physically necessary. The last one relates the orthogonality between states in each system to the ortho-complementation of L. It can be justified if one assumes that the orthogonality between states in the total system induces the ortho-complementation of L. We prove that if L satisfies these five conditions, then L is the separated product proposed by Aerts in 1982 to describe independent quantum systems (theorem 2). Finally, we give strong arguments to exclude the separated product and therefore our last condition. As a consequence, we ask whether among the ca-lattices that satisfy our first four basic necessary conditions, there exists an ortho-complemented one different from the separated product.Comment: Reports on Mathematical Physics, Vol. 50 no. 2 (2002), p. 155-16